another polarity riddle

forgive me for raising this venerable topic again, but...

i recently bought a pair of audiopax 88s to power my avantgarde duo 2.0s. the audiopax amps invert absolute polarity, so i have attached + to - on the taps of each amp. so far so good. the crossover of the AGs apparently also inverts polarity, so that it i recommended that the jumper is attached + to - from the mid horn to the treble horn. now i'm confused. wont following this advice negate the polarity shift from the amps -- at least as to the treble horn -- or does the crossover in the mid horn take care of this. thanks for any guidance.
Two inversions of polarity (phase) are the same as none--180 degrees out of phase twice is 360 degrees out of phase is 0 degrees out of phase. So, in some sense they do cancel each other out, but that is OK. Frankly, happens inside your equipment all the time as it passes through various amplification stages/op amps/whatever. As long as polarity is inverted an even number of times, the device or system doesn't invert absolute polarity. If it does it an odd number of times, it does invert polarity.

So, if I understand right, you have an amp that inverts polarity, so your solution of + to - establishes correct absolute polarity at the speaker ends of the cables. That appears to be correct for the mid-horn. Then, you have a crossover that inverts polarity, so the jumpers get switched again, reestablishing correct polarity to the treble horns (assuming speaker wires are attached to the + and - of the mid-horn terminals).

The alternative, which should work just as well, is to maintain inverted polarity at the ends of your speaker cables (i.e., connect + to + and - to - at the amp end), run the inverted output to the treble horn terminal (which will then re-invert to correct absolute phase), and jumper the treble horn terminal to the mid-horn terminal crossing + and - (thus reinverting the inverted phase at the end of the speaker cables).

The good news is that most people can't actually tell the difference between absolute phase and inverted phase anyway, and that half your source material (CDs, records) probably have recordings that don't preserve absolute phase anyway.

I believe you would hear a difference if two speakers in a box were out-of-phase with respect to each other, which means I think its freakishly odd that your speaker manufacturer didn't do the simple thing of wiring the terminals inside the speaker to correspond to consistent absolute phase. Asking end users to jumper the speakers in reverse seems just silly to me. And a way of getting bad reviews, since 90% of the world isn't going to read the instructions anyway.

Are you really sure the treble and mid are invert phase with respect to each other?
thank for the detailed response. yes, the avantgarde duo 2.0 inverts polarity in the crossover to the treble horn. the newer omega upgrade features a crossover that maintains polarity. dont ask me why they would do it that way. the good news is that i have followed your advice and it sounds fantastic. i apprecite the help.
If you have access to a spectrum analyser, when playing white noise it will show a deep notch at the crossover frequency if the drivers are out of phase.
My Jadis amp inverts polarity - the manual asks to switch + and - in the speakercable at the amp side to solve the issue - now my AG UNOs invert polarity again, so I decided to keep the cables at the amp side with + and - correctly attached. I do notice a better resolution and laid back presentation with this configuration rather than than following the continuous polarity changes along my setup.